Central Limit Theorems for Non-invertible Measure Preserving Maps

This paper is motivated by the question “How can we produce the characteristics of a Wiener process (Brownian motion) from a semi-dynamical system?”. This question is intimately connected with Central Limit Theorems for non-invertible maps and various invariance principles. Many results on CLT and invariance principles for maps have been proved, see e.g. the surveys Denker [4] and Mackey and Tyran-Kamińska [16]. These results extend back over some decades, and include the work of Boyarsky and Scarowsky [3], Gouëzel [7], Jab loński and Malczak [11], Rousseau-Egele [24], and Wong [31] for the special case of maps of the unit interval. Martingale approximations, developed by Gordin [6], were used by Keller [12], Liverani [15], Melbourne and Nicol [18], Melbourne and Török [19], and Tyran-Kamińska [26], to give more general results. Throughout this paper, (Y,B, ν) denotes a probability measure space and T : Y → Y a non-invertible measure preserving transformation. Thus ν is invariant under T i.e. ν(T−1(A)) = ν(A) for all A ∈ B. The transfer operator PT : L1(Y,B, ν) → L1(Y,B, ν), by definition, satisfies